Boundary Nevanlinna�CPick interpolation via reduction and augmentation

We give an elementary proof of Sarason??s solvability criterion for the Nevanlinna?CPick problem with boundary interpolation nodes and boundary target values. We also give a concrete parametrization of all solutions of such a problem. The proofs are based on a reduction method due to Julia and Nevanlinna. Reduction of functions corresponds to Schur complementation of the corresponding Pick matrices.     

基于分支割平面的一类无容量限制设施选址问题求解算法

无容量限制设施选址问题是经典的组合优化问题, 具有广泛的应用价值,然而该问题已被证明是NP难问题, 并且传统的分支定界方法求解速度较慢.研究以最大化总收益费用与总投建费用之差为目标的无容量限制设施选址问题,将其转化为节点包装问题,并根据模型的图形特点提出了新的合法不等式族------轴不等式族,经过严格的数学证明后得出轴不等式要强于原有的奇洞不等式. 同时,设计出切割不等式快速搜索算法嵌入到分支割平面方法中. 最后,通过实验验证了轴不等式族的强有效性, 以及分支割平面方法比分支定界方法求解速度快、节点数量少的优点.     

A case study in program transformation

A simple problem relating to binary trees is considered, and a recursive program is proved to solve this problem. The program may be transformed into a more efficient iterative one, usinggoto-statements and explicit stack handling. An independent correctness proof for this transformed version is given and compared with the original proof. Finally it is shown how a non-recursive conception of the problem may yield a structured iterative program, based onwhile-statements. A proof of this program is presented, reflecting its non-recursive conception, and it turns out that this proof has little similarity with the other two.     

The number of tournaments with a unique spanning cycle

The number of tournaments T_n on n nodes with a unique spanning cycle is the (2n-6)th Fibonacci number when n ≥ 4. Another proof of this result is given based on a recursive construction of these tournaments.     

The mühlbach-neville-aitken algorithm and some extensions

A new proof of the Mühlbach-Neville-Aitken algorithm for interpolation by a linear family of functions forming a Chebyshev system is given. This proof is based on Sylvester's identity for determinants. The algorithm is then applied to the general interpolation problem, and applications to orthogonal polynomials and Padé-type approximants are treated. Finally the extension to rational interpolation is also studied.     

On an identity of Hilbert

A simple proof of the polynomial identity used by Hilbert in the solution of the Waring problem is given. The proof is based on the continued fraction expansion of a certain formal hypergeometric series. Translated fromMatematischeskie Zametki, Vol. 66, No. 4, pp. 527–532, October, 1999.     

Convergence of Gaussian Quadrature and Lagrange Interpolation in Haar Systems

Consider a Markov system of functions whose linear span is dense with respect to the uniform norm in the space of the continuous functions on a finite interval. Gaussian rules are those which correctly integrate as many successive basis functions as possible with the lesser number of nodes. In this paper we provide a simple proof of the fact that such rules converge for all bounded Riemann-Stieltjes integrable functions. The proposed proof is also valid for any sequence of quadrature rules with positive coefficients which converge for the basis functions. Taking the nodes of the Gaussian rules as nodes for Lagrange interpolation, we give a sufficient condition for the convergence in L ₂-norm of such processes for bounded Riemann-Stieltjes integrable functions.     

On the Alexander-Hirschowitz theorem

The Alexander-Hirschowitz theorem says that a general collection of k double points in imposes independent conditions on homogeneous polynomials of degree d with a well-known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a self-contained proof based mainly on the previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case d=3, where our proof is shorter. We end with an account of the history of the work on this problem.     

Solving the 2-rooted mini-max spanning forest problem by branch-and-bound

The 2-rooted mini-max spanning forest problem is to find a spanning forest with two given root nodes on an undirected graph, such that the maximum tree cost in this forest is minimized. We introduce a branch-and-bound algorithm based on selecting nodes. On test instances up to 50 nodes the algorithm gives much better computational results than a known algorithm that is based on selecting edges. Furthermore the new algorithm can easily solve problem instances up to 80 nodes.     

Matrix equation, matrix polynomial and the number of zeros of a polynomial inside the unit circle

Using the idea of controllability, we present an alternative proof of a known result on the solution of the classical Schur-Cohn problem. The proof is based on an unit circle theorem derived in this paper.     

An eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities

A multiplicity result for an eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities is obtained. The proof is based on a three critical points theorem for nondifferentiable functionals.     

On Initial Data of a Simple Conservative Scattering System That Can Be Transferred to Zero by a Sequence of Inputs from <Emphasis Type="Italic">l</Emphasis><Superscript>2</Superscript>

We describe a lineal of initial data of a simple conservative scattering system that can be transferred to zero by a sequence from l ². The proof is based on the known relationship between the Lax-Phillips scattering theory and the theory of unitary operator nodes developed by Szokefalvi-Nagy, Foias, and Brodskii. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 559–563, April, 2005.     

Optimization of passive optical network planning

Network planning problem typically involves large capital investment and can be formulated as an optimization problem where the objective is minimization of the first installed cost. We consider a passive optical network (PON) planning problem based on a residential area in Adana (Turkey). There are four possible primary node locations, twenty possible secondary node locations, and twenty-eight customers. We use genetic algorithm and mathematical modeling techniques to optimize the position of the primary and secondary nodes, their split levels and assigning customers to secondary nodes and secondary nodes to primary nodes under some constraints such as system’s attenuations and technical characteristics of all the equipment.     

Convergence and existence results for continuous- and discrete-time Riccati equations

We give a new proof for the convergence of the solution of a terminal value problem for the periodic Riccati differential equation towards its strong solution as t → ?∞. The proof is mainly based on well-known comparison results and also on an explicit representation formula for the solution that reflects precisely the dependence on the terminal value. Moreover, we give sufficient conditions for the existence of a periodic solution of the differential equation. Similar results are derived for the discrete-time Riccati equation.     

The Optimal Feedback Control Problem for Voigt Model with Variable Density

The work is devoted to the study of weak solvability of the optimal feedback control problem for the Voigt model with variable density. The proof is based on the approximation-topological approach. First, the solvability of the approximation problem is proved, and then, based on a priori estimates that are independent of the approximation parameter, it is shown that from the sequence of solutions of the approximation problem, a subsequence converging to the solution of the original problem can be chosen. Then it is shown that among weak solutions to the problem there is at least one that gives a minimum to a given cost functional.

     

SPT optimality (mostly) via linear programming

One of the oldest results in scheduling theory is that the Shortest Processing Time (SPT) rule finds an optimal solution to the problem of scheduling jobs on identical parallel machines to minimize average job completion times. We present a new proof of correctness of SPT based on linear programming (LP). Our proof relies on a generalization of a single-machine result that yields an equivalence between two scheduling problems. We first identify and solve an appropriate variant of our problem, then map its solutions to solutions for our original problem to establish SPT optimality. Geometric insights used therein may find further uses; we demonstrate two applications of the same principle in generalized settings.     

A special case of de Branges' theorem on the inverse monodromy problem

We give a new proof of a special case of de Branges' theorem on the inverse monodromy problem: when an associated Riemann surface is of Widom type with Direct Cauchy Theorem. The proof is based on our previous result (with M.Sodin) on infinite dimensional Jacobi inversion and on Levin's uniqueness theorem for conformal maps onto comb-like domains. Although in this way we can not prove de Branges' Theorem in full generality, our proof is rather constructive and may lead to a multi-dimensional generalization. It could also shed light on the structure of invariant subspaces of Hardy spaces on Riemann surfaces of infinite genus.This work was supported by the Austrian Founds zur Förderung der wissenschaftlichen Forschung, project-number P12985-TEC     

Special-valued l-groups and Abelian covers

This paper presents a new and independent proof of the theorem (proven first by Kopytov and Gurchenkov [7] and again by Reilly [10]) that covers of the Abelian l-variety are either representable or are Scrimger covers. The proof in this paper is based upon the l-Cauchy constructions of Ball [1]; once these are applied to the problem, the proof becomes elementary.     

A new proof of the rigidity problem

In this short note we give a new proof of the boundary rigidity problem in a Euclidean setting proved by Croke. Our method is based on the differentiability of Busemann functions and the characteristic of Euclidean metric on Riemannian manifolds without conjugate points.

     

Boundary-value problem for degenerate parabolic equation of high order with varying time direction

In the present paper we investigate a boundary-value problem for a forward-backward parabolic equation in a rectangular domain and prove the existence of unique regular solution to this problem. The proof of the uniqueness of the solution is based on the spectral method, and in the proof of existence of solution we use the method of separation of variables. In the introduction we give a survey of related works.